Divide and Conquer: Recursive Likelihood Function Integration for Hidden Markov Models with Continuous Latent Variables

• 成果类型: Article
• 署名作者: Reich, Gregor
• 署名单位: University of Zurich
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2018.1750
• 发表日期: 2018
• 页码: 1457-1470
• 关键词: discrete-choice models; maximum-likelihood; replacement

摘要

This paper develops a method to efficiently estimate hidden Markov models with continuous latent variables using maximum likelihood estimation. To evaluate the (marginal) likelihood function, I decompose the integral over the unobserved state variables into a series of lower dimensional integrals, and recursively approximate them using numerical quadrature and interpolation. I show that this procedure has very favorable numerical properties: First, the computational complexity grows linearly in the number of periods, making the integration over hundreds and thousands of periods feasible. Second, I prove that the numerical error accumulates sublinearly in the number of time periods integrated, so the total error can be well controlled for a very large number of periods using, for example, Gaussian quadrature and Chebyshev polynomials. I apply this method to the bus engine replacement model of Rust [Econometrica 55(5): 999-1033] to verify the accuracy and speed of the procedure in both actual and simulated data sets.